Recombinant EphB4-HSA Fusion Protein With Standard Chemotherapy Regimens in Treating Patients With Advanced or Metastatic Solid Tumors

Inclusion Criteria: * Patients must have an Eastern Cooperative Oncology Group (ECOG) performance status of 0 or 1 * Patients must have measurable disease per Response Evaluation Criteria in Solid Tumors (RECIST) version (v)1.1 * Patients must have a life expectancy of at least 12 weeks * Patients must be able to comprehend and provide written informed consent * Women of childbearing potential (WOCBP) and male patients with WOCBP partner must be using an adequate method of contraception to avoid pregnancy throughout the study and for up to 12 weeks after the last dose of investigational product in such a manner that the risk of pregnancy is minimized; WOCBP include any female who has experienced menarche and who has not undergone successful surgical sterilization (hysterectomy, bilateral tubal ligation, or bilateral oophorectomy) or is not postmenopausal; post menopause is defined as: * Amenorrhea \>= 12 consecutive months without another cause or * For women with irregular menstrual periods and on hormone replacement therapy (HRT), a documented serum follicle stimulating hormone (FSH) level \> 35 mIU/mL * Women who are using oral contraceptives, other hormonal contraceptives (vagina products, skin patches, or implanted or injectable products), or mechanical products such as an intrauterine device or barrier methods (diaphragm, condoms, spermicides) to prevent pregnancy, or are practicing abstinence or where their partner is sterile (e.g., vasectomy) should be considered to be of childbearing potential * WOCBP must have a negative serum test (minimum sensitivity 25 IU/L or equivalent units of human chorionic gonadotropin \[HCG\]) within 72 hours prior to the start of investigational product * Patients with hepatitis B infection must be on appropriate antiviral therapy * ARM A COHORT 1: Patients must have advanced pancreatic adenocarcinoma (unresectable or metastatic) * ARM A COHORT 1: Patients must not have received prior therapy for metastatic or advanced disease; adjuvant therapy that is gemcitabine based is allowed as long as the adjuvant treatment was completed \>= 6 months before the diagnosis of recurrent disease * ARM A COHORT 1: Absolute neutrophil count \>= 1,500/ul * ARM A COHORT 1: Platelet count \>= 100,000/ul * ARM A COHORT 1: Aspartate aminotransferase (AST) and alanine aminotransferase (ALT) =\< 2 x upper limit of normal (ULN); if liver metastases are present then must be \< 5 X the ULN * ARM A COHORT 1: Total bilirubin =\< 1.5 x ULN * ARM A COHORT 1: Creatinine =\< 1.5 x institutional ULN, or creatinine clearance \>= 50 mL/min (calculated with the Cockcroft-Gault formula) * ARM A COHORT 1: Serum albumin \>= 2.5 g/dL * ARM B COHORT 2: Patients must have a histologically confirmed diagnosis of head and neck (squamous cell) carcinoma * ARM B COHORT 2: Patients must have failed treatment with platinum based therapy with or without cetuximab; previous therapy with a platinum concurrent with radiation followed by progression of disease within 6 months will count as failure of one prior line of cisplatin based therapy * ARM B COHORT 2: Absolute neutrophil count \>= 1,500/ul * ARM B COHORT 2: Platelet count \>= 100,000/ul * ARM B COHORT 2: AST and ALT =\< 2 X ULN; if liver metastases are present then must be \< 5 X the ULN * ARM B COHORT 2: Total bilirubin =\< 1.5 x institutional upper limit of normal (IULN) * ARM B COHORT 2: Serum albumin \>= 2.5 g/dL * ARM B COHORT 2: Creatinine =\< 1.5 x institutional ULN, or creatinine clearance \>= 50 mL/min (calculated with the Cockcroft-Gault formula) * ARM B COHORT 3: Patients must have a histologically confirmed diagnosis of non-small cell lung cancer * ARM B COHORT 3: Patients must have failed one prior line of systemic therapy for the treatment of advanced non-small cell lung cancer; prior adjuvant therapy with subsequent recurrence within 6 months of completion of therapy will count as one prior line of systemic therapy and such patients will be eligible * ARM B COHORT 3: Absolute neutrophil count \>= 1,500/ul * ARM B COHORT 3: Platelet count \>= 100,000/ul * ARM B COHORT 3: AST and ALT =\< 2 X ULN; if liver metastases are present then must be \< 5 X the ULN * ARM B COHORT 3: Total bilirubin =\< 1.5 x IULN * ARM B COHORT 3: Serum albumin \>= 2.5 g/dL * ARM B COHORT 3: Creatinine =\< 1.5 x institutional ULN, or creatinine clearance \>= 50 mL/min (calculated with the Cockcroft-Gault formula) * ARM C COHORT 4: Patients must have a histologically or cytologically proven diagnosis of advanced (unresectable or metastatic) gallbladder cancer or cholangiocarcinoma and be candidates for first line therapy with gemcitabine and cisplatin * ARM C COHORT 4: Patients must not have received prior systemic chemotherapy for advanced or metastatic disease; prior adjuvant chemotherapy or concurrent chemotherapy and radiation are allowed if they were completed \>= 6 months prior to the diagnosis of recurrent disease * ARM C COHORT 4: Absolute neutrophil count \>= 1,500/ul * ARM C COHORT 4: Platelet count \>= 100,000/ul * ARM C COHORT 4: AST and ALT =\< 5 x ULN * ARM C COHORT 4: Total bilirubin =\< 2 X IULN * ARM C COHORT 4: Serum albumin \>= 2.5 g/dL * ARM C COHORT 4: Creatinine =\< 1.5 x institutional ULN, or creatinine clearance \>= 50 mL/min (calculated with the Cockcroft-Gault formula) * ARM C COHORT 4: Patients who have had decompression of the biliary tree within the last 14 days, must have a stable bilirubin level as confirmed by two measurements that are within 5 to 7 days of each other; (the second measurement must be obtained within 7 days prior to registration); both the first and second measurement must be =\< 2 X IULN; stability is defined as the second measurement being no more than one point higher than the first Exclusion Criteria: * Patients must not have received any anti-cancer therapy (cytotoxic chemotherapy, targeted therapy or radiation) within the past 28 days prior to initiation of study therapy * Patients must not have an active major systemic infection requiring systemic antibiotics 72 hours or less prior to the first dose of study drug * Patients must not have untreated central nervous system (CNS) metastases; patients whose CNS metastases have been treated by surgery or radiotherapy, who are no longer on corticosteroids, and who are neurologically stable may be enrolled * Patients must not have any of the following: New York Heart Association (NYHA) class 3 or 4 congestive heart failure; myocardial infarction within the past 12 months, acute coronary syndrome, diabetes mellitus with ketoacidosis or chronic obstructive pulmonary disease (COPD) requiring hospitalization in the preceding 6 months; or any other intercurrent medical condition that contra-indicates treatment with sEphB4HSA or places the patient at undue risk for treatment related complications * Patients must not have any other condition, including mental illness or substance abuse, deemed by the Investigator to be likely to interfere with a patient's ability to sign informed consent, cooperate and participate in the study, or interferes with the interpretation of the results * Patients must not be pregnant of lactating * Patients must not be on any dose of warfarin or are on full dose anticoagulation with other agents, including low molecular weight heparin, antithrombin agents, anti-platelet agents and full dose aspirin within 7 days prior to first dose of study drug; patients on prophylactic doses of low-molecular weight heparin are allowed * Patients must not have had any active bleeding in the last =\< 4 weeks or have an otherwise known bleeding diathesis * Patients must not have corrected QT (QTc)F (Fridericia Correction Formula) \> 480 on 2 out of 3 electrocardiograms (EKG's) (if first EKG is =\< 480, no need to repeat, if first EKG is \> 480 repeat twice for a total of 3 EKG's) * Patients must not have uncontrolled hypertension as defined by systolic blood pressure (SBP) \>= 160 mmHg or diastolic blood pressure (DBP) \>= 90 mmHg; patients whose blood pressure can be controlled medically are allowed to be rescreened once blood pressure (BP) is under control * Patients must not have \> grade sensory neuropathy * Patients must not have known human immunodeficiency virus (HIV) infection * ARM A COHORT 1: Patients must not have prior nab-paclitaxel exposure * ARM A COHORT 1: Patients must not have a history of slowly progressive dyspnea and unproductive cough, or of conditions such as sarcoidosis, silicosis, idiopathic pulmonary fibrosis, pulmonary hypersensitivity pneumonitis, or multiple allergies * ARM B COHORT 2: Patients must not have prior exposure to docetaxel * ARM B COHORT 2: Patients with head and neck cancer must not have radiologic evidence of major arterial involvement * ARM B COHORT 3: Patients must not have prior exposure to docetaxel * ARM B COHORT 3: Patients must not have had more than one prior line of systemic therapy for advanced non-small cell lung cancer (including treatment with a targeted agent); prior adjuvant therapy completed more than 6 months prior to disease recurrence will not count as a prior line of systemic therapy for advanced disease * ARM B COHORT 3: Patients must not have active clinically significant hemoptysis * ARM B COHORT 3: Patients must not have a central lesion with radiologic evidence of arterial involvement * ARM C COHORT 4: Patients must not have prior cisplatin exposure